I have seen the Eye of Horus symbol represented as a left eye in some places and as right eye in other places.

I am wondering are they simply different orientations of the same symbol or do they each carry a different symbolic meaning?

There are actually two Eyes in Egyptian mythology. One is the Eye of Ra, which is mythically the right eye of the god, and delivers burning heat to its enemies. The right eye of Ra (or often attributed to Horus as well) is called the /ir.t/ (iret), or "doer" eye, and is mythically associated with several goddesses who protect the sun-god by fiercely fighting away his enemies in the firm of his "Fiery Eye", whcih is often personified as a uraeus cobra (Roberts 1995). Sekhmet, for example, is such a Fiery Eye goddess, who blasts Ra's enemies during the nightly journey of the sun-god into the underworld. The myth of the Destruction of Mankind gives an example of the fierceness of these goddesses when called upon to do Ra's bidding. Otehr goddesses associated with being a Fiery Eye goddess in Ra's command is Isis, Ma'at, Neith, Bastet, and so on.

Rarely used as an amulet, it does appear periodically mainly as a means to ward off one's enemies, as the power of the /ir.t/ goddesses are said to protect the wearer.

Meanwhile, in the Contendings of Seth and Horus, it's said that the left eye of Horus was damaged in his fight with his uncle Seth (Sutekh), which was cured by either Thoth or Isis (depending upon which version you read). However, as Horus is also a sun deity, it was said that his left eye was never as strong as his right again, which is why the left eye of the sun was thought to be the moon.

Because Horus's eye was damaged and then made whole, it is referred to as the /wAD.t/ (wadjet) eye, or the eye which was "made whole/repaired." It is usually the wadjet eye one sees as amulets for wearing around the neck as a means to "make whole" the person fro any ailment he has.